Fading Channels, Diversity and MIMO systems · Fading Channel Summary • In general, this is...

© Keith M. Chugg, 2017

Fading Channels, Diversity and MIMO systems

EE564: Digital Communication and Coding Systems

Keith M. ChuggSpring 2017

1


Overview Topics

• Fading channel models

• Performance impact of fading

• Benefits of diversity

• Methods for obtaining diversity

• MIMO systems

• Space time codes (for diversity)

• Space time multiplexing (for increased throughput)

• Capacity measures for MIMO and fading channels

2

© Keith M. Chugg, 2017 3

Mobile Communication Systems c⃝Keith M. Chugg, USC – August 1999

Typical 3-Level Channel Models

• Path Loss

– Deterministic propagation loss model

– Large scale

– Empirically determined from field measurements

• Shadowing

– Statistical model for the deviation from the path loss model

– Long-term fading – e.g., 10-100 wavelengths

– Empirically determined from field measurements

• Fading

– Statistical model for short-term (sub-wavelength) power fluctuations

– Also characterizes the distortion characteristics of the channel

– Simple analytical models, verified via measurements

17


Shadowing deviation

20 wavelengths

Received Power (dB)

Distance from the transmitter

-40

-30

-20

-10

0

10

Long-term average set by shadow level


Relation Between Three Levels of Channel Models

18


h1

h2

h1, h2 ≪ d

d


Path Loss Models

• Free Space:Pr(d)

Pr(d0)=

⎛

⎜

⎝

d

d0

⎞

⎟

⎠

−2

– Power spread evenly over sphere of radius d

• Single Ground Reflection:

Pr(d)

Pr(d0)=

⎛

⎜

⎝

d

d0

⎞

⎟

⎠

−4

19


Ellipse of constant delay: Tx and Rx at foci

TxRx


Path Loss Models

• Multipath Reflection Environments:

Pr(d)

Pr(d0)=

⎛

⎜

⎝

d

d0

⎞

⎟

⎠

−β

⎡

⎢

⎢

⎣

Pr(d)

Pr(d0)

⎤

⎥

⎥

⎦

dB

= −10β log10

⎛

⎜

⎝

d

d0

⎞

⎟

⎠

– β is the path loss exponent

∗ Typical macrocellular: β ∼ 3 to 4∗ Typical microcellular: β ∼ 2 to 8

20



Path Loss Models

• Models are Roughly Frequency Independent

– Weak dependency described in more detailed model

– More difficult to predict in smaller regions (e.g., indoor)

– Environment specific models: ray-tracing, Manhattan pico cells, etc.

• Power decays linearly (in dB) with delay

– Free space ⇒ 20 dB per decade

– β ⇒ 10β dB per decade

• Utility of path loss models:

– rough cell planning (e.g., cell size, reuse factors)

21



Shadowing Models

• Random deviation from path loss model:Pr,S(d; u)

Pr(d0)= ϵ(u)

Pr(d)

Pr(d0)⎡

⎢

⎢

⎣

Pr,S(d; u)

Pr(d0)

⎤

⎥

⎥

⎦

dB

=

⎡

⎢

⎢

⎣

Pr(d)

Pr(d0)

⎤

⎥

⎥

⎦

dB

+ 10 log10 [ϵ(u)]

= −10β log10

⎛

⎜

⎝

d

d0

⎞

⎟

⎠ + ϵdB(u)

• Common Model: Log-Normal Shadowing

ϵdB(u) ∼ N (·; 0;σ2ϵdB

)

– The received power in dB may be thought of as Gaussian with meangiven by the path loss model and variance σ2

ϵdB

• Shadowing deviation: σϵdB

– Macrocellular systems have values in the range 5 to 12, with 8 beingtypical

22


t

h(t1; τ)

h(t2; τ )h(t3; τ )

h(t4; τ )

τ


Short-term (multipath) Fading Models

• Common Model: random, time-varying linear system

– Impulse response from a delta applied at time t is h(u; t; τ )

y(u, τ ) = h(u; t; τ ) ∗ x(τ ) z(u, τ ) = h(u; t + δ; τ ) ∗ x(τ )

z(u, τ ) = y(u, τ )

24




• Characterizing Distortion: What is the shape of the impulseresponse h(u; t; τ ) wrt τ?

– τd: Delay Spread – how long does the channel ring from a timeimpulse?

– Bc: Coherence Bandwidth – over what range of frequencies is thegain of the channel flat?

• Characterizing Time-variation: How does h(u; t; τ ) change witht?

– tc: Coherence time – for what value of ∆ are the responses at t andt +∆ uncorrelated?

– fd: Doppler Spread – how much will the spectrum of an input tone(i.e., frequency impulse) be spread in frequency?

25


variation in t variation inτ

Time-variation Properties

Distortion Properties

Time Domain

Frequency Domain

Coherence Time

Doppler Spread

Delay Spread

Coherence Bandwidth



• Distortion Properties: Bc ∝ 1τd

• Time-variation Properties: fd ∝ 1tc

26



Measures Relative to Signals

• Does the channel distort the signal?

– W ≪ Bc ⇒ NO ⇒ Flat Fading

– W ≥ Bc ⇒ YES ⇒ Frequency-Selective Fading

∗ Note: If W ∼= 1T , then frequency selective fading implies that

T ≤ τd ⇒ time dispersion or intersymbol interference (ISI)∗ Not so for wideband systems – W ≫ 1

T

∗ Flat Fading ⇐⇒ amplitude and phase distortion only!

• Does the channel remain constant over many channeluses?

– T ≪ tc ⇒ YES ⇒ Slow Fading

– T ≥ tc ⇒ NO ⇒ Fast Fading

∗ Slow fading may still require frequent training and/or adaptivetracking

31



Clarke’s Doppler Model: Meaning (flat fading)

• I/Q carrier modulated inputs:

x(t) = xI(t)√

2 cos(2πfct) − xQ(t)√

2 sin(2πfct)

= ℜ!

x(t)√

2ej2πfct"

= |x(t)| cos(2πfct + x(t))

x(t) = xI(t) + jxQ(t)

• Output:

y(u; t) = [hI(t)xI(t) − hQ(t)xQ(t)]√

2 cos(2πfct)

−[hI(t)xQ(t) + hQ(t)xI(t)]√

2 sin(2πfct)

= ℜ!

y(t)√

2ej2πfct"

= |y(t)| cos(2πfct + y(t))

y(t) = yI(t) + jyQ(t) = x(t)h(t)

36


-50

-40

-30

-20

-10

0

10

2000 2500 3000 3500 4000

(a)(b)F

adin

g A

mpl

itud

e (d

B)

time


Power in Sample Realizations

38

this is the envelope for Rayleigh (flat) fading


Fading Channel Summary

• In general, this is complex stuff…

• Many modern systems use OFDM, so the sub-carrier channels are modeled as frequency flat fading.

• Correlation in complex gains across frequencies, several coherence bandwidth in a broadband OFDM system

• Rayleigh fading is worse case: I and Q channel gains are zero mean, independent Gaussian. Results from many, many diffuse scatters

• Ricean fading is similar with non-zero means in the I and Q channel gains

• Time variation is often modeled as

• Fixed or quasi-static

15



Effects of Fading

• Recall: for the AWGN channel, for all modulations considered, theerror performance decays exponentially in SNR

Pb∼= K1e

−K2EbN0

• Fading:

– Random variations in received power

– Average the AWGN performance over the statistics Eb/N0

– Consider the performance as a function of average Eb/N0

– Performance decays only inverse linearly with Rayleigh (flat) fading

Pb∼= K

⎡

⎢

⎣

Eb

N0

⎤

⎥

⎦

−1

65


10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

-10 0 10 20 30 40 50

Ps(4)Ps(8)Ps(16)

Ps(

M)

- Sy

mbo

l Err

or R

ate

Es/No (dB)

Raleigh FadingNo Fading


Effects of Fading – PSK

• Intuition: worst case dominates!

α10−1 + (1 − α)10−6 ∼= α10−1 ≫ 10−6

66



Combating Fading: Diversity

• Intuition: combining multiple independent copies of the receivedsignal will reduce the variance of the SNR

r(d)(t) = h(d)s(t; a) + n(d)(t) d = 1, 2 . . . D

– Diversity Order: D – number of effectively independent replicas

– Impact on Performance: Increases BER decay

Pb∼= K

⎡

⎢

⎣

Eb

N0

⎤

⎥

⎦

−D

– As D increases, the performance approaches that of no-fading!

67



How to Obtain Diversity

• Spatial Diversity:

– e.g., Space two antennas farther than λ/2 in dense scattering

• Time Diversity:

– e.g., Repeat the transmission after waiting longer than thecoherence time

• Frequency Diversity:

– e.g., Transmit the signal on two carriers spaced further than thecoherence BW

• Which type if best?

– Performance gains are the same regardless (nominally)

– Effort required to combine the diversity effectively may differ greatlywith the type and the exact signal format

68


-50

-40

-30

-20

-10

0

10

2000 2500 3000 3500 4000

(a)(b)F

adin

g A

mpl

itud

e (d

B)

time


Intuitive View of Diversity

69



Optimal Diversity Combining

• Optimal Digital Communication Receiver:

– Consider all possible versions of the received signal (includingdistortion, interference, etc.) that arise from possible a

– Correlate with each of these possibilities

– Adjust correlation for energy difference

– Maximize over possibilities

• This yields Maximum (Signal-to-Noise) Ratio Combining:

zd(a) =!

r(d)(t)s(t; a)dt

Z(a) =D"

d=1

#

h(d)$∗

zd(a)

– If each signal s(t; a) has equal energy, then

maxa

Z(a)

70


Performance of BPSK in Rayleigh Fading

Rayleigh fading (no diversity)

22

� = Eb/N0 = random due to fading with mean � = Eb/N0


f(�) =1

�e��/� � > 0


f(�) =1

�e��/� � > 0

P (E) =Z 1

0Q(

p2�)f(�)d�

=

1

2

1�

r�

1 + �

�

⇡ 1

4�� 1

Note: all mods we have seen have uncoded performance that is well approximated as a Q-function


Performance of BPSK in Rayleigh Fading

Rayleigh fading, diversity D and MRC combining

23



f(�) =1

�e��/� � > 0

P (E) =Z 1

0Q(

p2�)f(�)d�

=

1

2

1�

r�

1 + �

�

⇡ 1

4�� 1

f(�) =1

(D � 1)!�D�D�1e��/� � > 0

Central chi-squared with 2D degrees of freedom


f(�) =1

�e��/� � > 0

P (E) =Z 1

0Q(

p2�)f(�)d�

=

1

2

1�

r�

1 + �

�

⇡ 1

4�� 1

f(�) =1

(D � 1)!�D�D�1e��/� � > 0

P (E) =Z 1

0Q(

p2�)f(�)d�

=

✓1� µ

2

◆D D�1Y

k=0

⇣L� 1 + k

k

⌘✓1 + µ

2

◆k

µ =

r�

1 + �

P (E) ⇡⇣

2D � 1

D

⌘✓1

4�

◆D

� � 1


Original signal spectrumSpread spectrum

Channel Coherence BW

Noise floor


Practical Frequency Diversity: Spreading

• Use more bandwidth than required:

– provides frequency diversity ⇐⇒ frequency-selectivity

– spectrally inefficient (single-user)

• Techniques:

– Direct Sequence: mix with a pseudorandom squarewave carrier

– Frequency Hopping: change fc according to a pseudorandompattern

– Time Hopping: change signal epoch of narrow pulse inpseudorandom manner

77


Spread spectrum

Noise floor

Interference

Despread spectrum

Noise floor

Interference

Modulation and Coding Channel

DS Spread Spectrum


DS Spread Spectrum

78

D ~ number of coherence BWs in the spread BW


1/W

h∗0

1/W 1/W 1/W

h∗L+1 h∗

L h∗L−1

r(t)

RAKE is the Matched Filter to the FS Channel


DS Spread Spectrum

• Spreading Ratio: η = Tb/Tc; Tc = chip time

– Also called processing gain since an interferer’s in-band power isreduce by η−1 after despreading

• Frequency Diversity Combining: RAKE receiver

79


Error Correction

Code

Interleaver (scrambler) Modulator

Time-varying Fading Channel

Decoder Deinterleaver Demodulator

Spreads dependance/redundancy

in time


Practical Time Diversity: Interleaving and Coding

• Forward Error Correction Coding:

– Provides an SNR gain (i.e., coding gain) on AWGN channel

– Also provides (small) diversity gain on a time-varying fading channel

• Interleaving:

– Greatly improves the diversity gain associated with coding

– Useless without coding

72

D ~ number of coherence times in the code block


Practical Diversity

• In the above, we do not have access to parallel, decoupled diversity branches

• diversity is coupled together through the signaling

• general results still hold

• obtained by doing some form of whitening/decorrelation on the correlated fading metrics

28


MIMO Systems

29

TX RX...

.

.

.

Nt transmit antennas Nr receive antennas

Channel H

Channel H

x

w

z = Hx+w

This is a single channel use


MIMO Systems

• Typical Channel Model

• Each element of H is an independent, flat-fading, Rayleigh channel

• Space Time Codes (STCs):

• Use to get diversity against multi path fading

• Typically model the channel as not changing during code blocks

• Very short code blocks — these are really ST Modulations

• Space Time Multiplexing:

• Just send a different QASK signal over each TX antenna

• If Nt >= Nr, can support Nt “spatial streams”

30


Space Time Codes

31

Suggest basic design rules for STCs:Rank and Determinant Criterion

744 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 2, MARCH 1998

Space–Time Codes for High DataRate Wireless Communication:

Performance Criterion and Code ConstructionVahid Tarokh, Member, IEEE, Nambi Seshadri, Senior Member, IEEE, and A. R. Calderbank, Fellow, IEEE

Abstract—We consider the design of channel codes for im-proving the data rate and/or the reliability of communicationsover fading channels using multiple transmit antennas. Data isencoded by a channel code and the encoded data is split intostreams that are simultaneously transmitted using transmit

antennas. The received signal at each receive antenna is a linearsuperposition of the transmitted signals perturbed by noise. Wederive performance criteria for designing such codes under theassumption that the fading is slow and frequency nonselective.Performance is shown to be determined by matrices constructedfrom pairs of distinct code sequences. The minimum rank amongthese matrices quantifies the diversity gain, while the minimumdeterminant of these matrices quantifies the coding gain. Theresults are then extended to fast fading channels. The designcriteria are used to design trellis codes for high data rate wirelesscommunication. The encoding/decoding complexity of these codesis comparable to trellis codes employed in practice over Gaussianchannels. The codes constructed here provide the best tradeoffbetween data rate, diversity advantage, and trellis complexity.Simulation results are provided for 4 and 8 PSK signal setswith data rates of 2 and 3 bits/symbol, demonstrating excellentperformance that is within 2–3 dB of the outage capacity for thesechannels using only 64 state encoders.

Index Terms—Array processing, diversity, multiple transmitantennas, space–time codes, wireless communications.

I. INTRODUCTION

A. Motivation

CURRENT cellular standards support circuit data and faxservices at 9.6 kb/s and a packet data mode is being

standardized. Recently, there has been growing interest inproviding a broad range of services including wire-line voicequality and wireless data rates of about 64–128 kb/s (ISDN)using the cellular (850-MHz) and PCS (1.9-GHz) spectra [2].Rapid growth in mobile computing is inspiring many proposalsfor even higher speed data services in the range of 144 kb/s(for microcellular wide-area high-mobility applications) andup to 2 Mb/s (for indoor applications) [1].The majority of the providers of PCS services have further

decided to deploy standards that have been developed atcellular frequencies such as CDMA (IS-95), TDMA (IS-54/IS-136), and GSM (DCS-1900). This has led to considerableManuscript received December 15, 1996; revised August 18, 1997. The

material in this paper was presented in part at the IEEE InternationalSymposium on Information Theory, Ulm, Germany, June 29–July 4, 1997.The authors are with the AT&T Labs–Research, Florham Park, NJ 07932

USA.Publisher Item Identifier S 0018-9448(98)00933-X.

effort in developing techniques to provide the aforementionednew services while maintaining some measure of backwardcompatibility. Needless to say, the design of these techniquesis a challenging task.Band-limited wireless channels are narrow pipes that do not

accommodate rapid flow of data. Deploying multiple transmitand receive antennas broadens this data pipe. Information the-ory [14], [35] provides measures of capacity, and the standardapproach to increasing data flow is linear processing at thereceiver [15], [44]. We will show that there is a substantialbenefit in merging signal processing at the receiver withcoding technique appropriate to multiple transmit antennas.In particular, the focus of this work is to propose a solutionto the problem of designing a physical layer (channel coding,modulation, diversity) that operate at bandwidth efficienciesthat are twice to four times as high as those of today’s systemsusing multiple transmit antennas.

B. DiversityUnlike the Gaussian channel, the wireless channel suffers

from attenuation due to destructive addition of multipaths inthe propagation media and due to interference from other users.Severe attenuation makes it impossible for the receiver todetermine the transmitted signal unless some less-attenuatedreplica of the transmitted signal is provided to the receiver.This resource is called diversity and it is the single mostimportant contributor to reliable wireless communications.Examples of diversity techniques are (but are not restricted to)• Temporal Diversity: Channel coding in conjunction withtime interleaving is used. Thus replicas of the transmit-ted signal are provided to the receiver in the form ofredundancy in temporal domain.

• Frequency Diversity: The fact that waves transmitted ondifferent frequencies induce different multipath structurein the propagation media is exploited. Thus replicas ofthe transmitted signal are provided to the receiver in theform of redundancy in the frequency domain.

• Antenna Diversity: Spatially separated or differently po-larized antennas are used. The replicas of transmittedsignal are provided to the receiver in the form of redun-dancy in spatial domain. This can be provided with nopenalty in bandwidth efficiency.

When possible, cellular systems should be designed to encom-pass all forms of diversity to ensure adequate performance

0018–9448/98$10.00 1998 IEEE

IEEE JOURNAL ON SELECT AREAS IN COMMUNICATIONS, VOL. 16, NO. 8, OCTOBER 1998 1451

A Simple Transmit Diversity Techniquefor Wireless Communications

Siavash M. Alamouti

Abstract— This paper presents a simple two-branch trans-mit diversity scheme. Using two transmit antennas and onereceive antenna the scheme provides the same diversity orderas maximal-ratio receiver combining (MRRC) with one transmitantenna, and two receive antennas. It is also shown that thescheme may easily be generalized to two transmit antennas and

receive antennas to provide a diversity order of 2 . Thenew scheme does not require any bandwidth expansion anyfeedback from the receiver to the transmitter and its computationcomplexity is similar to MRRC.

Index Terms—Antenna array processing, baseband processing,diversity, estimation and detection, fade mitigation, maximal-ratio combining, Rayleigh fading, smart antennas, space blockcoding, space–time coding, transmit diversity, wireless commu-nications.

I. INTRODUCTION

THE NEXT-generation wireless systems are required tohave high voice quality as compared to current cellular

mobile radio standards and provide high bit rate data ser-vices (up to 2 Mbits/s). At the same time, the remote unitsare supposed to be small lightweight pocket communicators.Furthermore, they are to operate reliably in different types ofenvironments: macro, micro, and picocellular; urban, subur-ban, and rural; indoor and outdoor. In other words, the nextgeneration systems are supposed to have better quality andcoverage, be more power and bandwidth efficient, and bedeployed in diverse environments. Yet the services must re-main affordable for widespread market acceptance. Inevitably,the new pocket communicators must remain relatively simple.Fortunately, however, the economy of scale may allow morecomplex base stations. In fact, it appears that base stationcomplexity may be the only plausible trade space for achievingthe requirements of next generation wireless systems.The fundamental phenomenon which makes reliable wire-

less transmission difficult is time-varying multipath fading [1].It is this phenomenon which makes tetherless transmission achallenge when compared to fiber, coaxial cable, line-of-sightmicrowave or even satellite transmissions.Increasing the quality or reducing the effective error rate in

a multipath fading channel is extremely difficult. In additivewhite Gaussian noise (AWGN), using typical modulation andcoding schemes, reducing the effective bit error rate (BER)from 10 to 10 may require only 1- or 2-dB higher signal-to-noise ratio (SNR). Achieving the same in a multipath fading

Manuscript received September 1, 1997; revised February 1, 1998.The author was with AT&T Wireless Services, Redmond, WA, USA. He is

currently with Cadence Design Systems, Alta Business Unit, Bellevue, WA98005-3016 USA (e-mail: [email protected]).Publisher Item Identifier S 0733-8716(98)07885-8.

environment, however, may require up to 10 dB improvementin SNR. The improvement in SNR may not be achieved byhigher transmit power or additional bandwidth, as it is contraryto the requirements of next generation systems. It is thereforecrucial to effectively combat or reduce the effect of fading atboth the remote units and the base stations, without additionalpower or any sacrifice in bandwidth.Theoretically, the most effective technique to mitigate mul-

tipath fading in a wireless channel is transmitter power control.If channel conditions as experienced by the receiver on oneside of the link are known at the transmitter on the other side,the transmitter can predistort the signal in order to overcomethe effect of the channel at the receiver. There are twofundamental problems with this approach. The major problemis the required transmitter dynamic range. For the transmitterto overcome a certain level of fading, it must increase its powerby that same level, which in most cases is not practical becauseof radiation power limitations and the size and cost of theamplifiers. The second problem is that the transmitter doesnot have any knowledge of the channel experienced by thereceiver except in systems where the uplink (remote to base)and downlink (base to remote) transmissions are carried overthe same frequency. Hence, the channel information has to befed back from the receiver to the transmitter, which resultsin throughput degradation and considerable added complexityto both the transmitter and the receiver. Moreover, in someapplications there may not be a link to feed back the channelinformation.Other effective techniques are time and frequency diversity.

Time interleaving, together with error correction coding, canprovide diversity improvement. The same holds for spreadspectrum. However, time interleaving results in large delayswhen the channel is slowly varying. Equivalently, spread spec-trum techniques are ineffective when the coherence bandwidthof the channel is larger than the spreading bandwidth or,equivalently, where there is relatively small delay spread inthe channel.In most scattering environments, antenna diversity is a

practical, effective and, hence, a widely applied techniquefor reducing the effect of multipath fading [1]. The classicalapproach is to use multiple antennas at the receiver andperform combining or selection and switching in order toimprove the quality of the received signal. The major problemwith using the receive diversity approach is the cost, size,and power of the remote units. The use of multiple antennasand radio frequency (RF) chains (or selection and switchingcircuits) makes the remote units larger and more expensive.As a result, diversity techniques have almost exclusively been

0733–8716/98$10.00 199 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 4, APRIL 1999 527

Signal Design for Transmitter DiversityWireless Communication SystemsOver Rayleigh Fading Channels

Jiann-Ching Guey, Michael P. Fitz, Mark R. Bell, and Wen-Yi Kuo, Member, IEEE

Abstract— In this paper, transmitter diversity wireless com-munication systems over Rayleigh fading channels using pilotsymbol assisted modulation (PSAM) are studied. Unlike conven-tional transmitter diversity systems with PSAM that estimatethe superimposed fading process, we are able to estimate eachindividual fading process corresponding to the multiple trans-mitters by using appropriately designed pilot symbol sequences.With such sequences, special coded modulation schemes can thenbe designed to access the diversity provided by the multipletransmitters without having to use an interleaver or expand thesignal bandwidth. The notion of code matrix is introduced forthe coded modulation scheme, and its design criteria are alsoestablished. In addition to the reduction in receiver complexity,simulation results are compared to, and shown to be superior to,that of an intentional frequency offset system over a wide rangeof system parameters.

Index Terms— Channel coding, diversity methods, Rayleighchannels.

I. INTRODUCTION

PROVIDING an architecture with diversity is importantfor maintaining high performance in wireless mobile

communications. Diversity can be achieved by using multipleantennas, using interleaved coded modulation, resolving prop-agation paths in time or spatially, and using multicarrier trans-mission [1], [2]. Perhaps the most commonly used technique isinterleaved coded modulation. The coding adds the redundancyto provide diversity and the interleaving separates the codesymbols to (hopefully) provide independent fading distortionfor each of the code symbols. The problem with standardinterleaved coded modulation is that a tradeoff must be madebetween decoding delay (a function of the interleaver depth)and demodulation performance. This is especially importantin applications where performance is decoding delay sensitive(e.g., voice transmission). For situations with small Dopplerspread (e.g., pedestrian or stopped vehicle), either a very long

Paper approved by P. H. Witke, the Editor for Communication Theory ofthe IEEE Communications Society. Manuscript received August 28, 1995;revised March 12, 1998 and July 27, 1998. This work was supported byPurdue Research Foundation and National Science Foundation under GrantNCR-9115820. This paper was presented in part at the 1996 IEEE VehicularTechnology Conference, Atlanta, GA.J.-C. Guey is with Ericsson Inc., Research Triangle Park, NC 27709 USA.M. P. Fitz is with the Department of Electrical Engineering, The Ohio State

University, Columbus, OH 43210-1272 USA.M. R. Bell is with the School of Electrical Engineering, Purdue University,

West Lafayette, IN 47907-1285 USA (e-mail: [email protected]).W.-Y. Kuo is with Lucent Technologies, Whippany, NJ 07981-0903 USA.Publisher Item Identifier S 0090-6778(99)03318-8.

interleaver is needed to achieve quasi-independent distortionon code symbols or else interleaving is not effective.An effective technique in wireless communications is trans-

mission diversity. The advantage of transmission diversity isthat by transmitting from multiple spatially separated antennas(e.g., a base station) diversity can be achieved without greatlyincreasing the complexity of the receiver (e.g., a portable unit).The simplest idea is to switch between the transmitters atdifferent time instants and allow only one transmitter to be onat a time. Because the transmitters are operated intermittently,their peak power is considerably higher than their averagepower, which complicates the design of their output amplifiers.Other transmission diversity techniques that do not switch offthe transmitter are ones using an intentional time offset [3] orfrequency offset [4], phase sweeping [5], frequency hopping[6], and modulation diversity [7]. Most of these techniquesuse phase or frequency modulation of each transmitter carrierto induce intentional time-varying fading at the receiver.1 Theadvantage of these techniques is that the modulation level ofthe carrier and the interleaving depth can be chosen to achievenear ideal interleaving. In these applications, a shorter inter-leaver depth is usually only achieved with an expanded signalbandwidth. The focus of this paper is the exposition of a fairlysimple alternate system architecture which can provide thediversity inherent in multiple transmissions without requiringinterleaving even with low mobility.In this paper we consider linear modulation on frequency

nonselective fading channels. Consequently, applications ofthis work are in modems using narrowband or multicarriermodulation. Decoding of error control codes in frequency non-selective fading channels requires an estimate of the channelstate (or multiplicative distortion), and transmitted referencetechniques usually provide the simplest method for channelstate estimation. Common transmitted reference techniquesare tone-calibration techniques (TCT) [8] and pilot symbolassisted modulation (PSAM) [9]. PSAM is preferred in prac-tice because it typically provides a better peak to averagetransmitted power ratio without the need to redesign themodulation pulse. Both TCT and PSAM are amenable to aperformance analysis for ideal interleaved coded modulation[1] and for correlated fading [10]. In fact, the work in [11]designed and analyzed the performance of a system usinginterleaved coded modulation and frequency offset diversity

1Note this is not the case for [3] and [7].

0090–6778/99$10.00 1999 IEEE

Very simple STC code for Nt = 2


Space Time (block) Codes

32

zk = Hxk +wk k = 0, 1, 2, . . . L� 1

Z = HX+W

The pairwise error probability, conditioned on H, will be:

zk = Hxk +wk k = 0, 1, 2, . . . L� 1

Z = HX+W

minx2C

kZ�HXk2

kH(Xi �Xj)k2

For receiver with CSI, the ML receiver iszk = Hxk +wk k = 0, 1, 2, . . . L� 1

Z = HX+W

minx2C

kZ�HXk2

PPW (Xi,Xj |H) = Q

0

@s

d2(i, j)

2N0

1

A

d2(i, j) = kH(Xi �Xj)k2

kH(Xi �Xj)k2

need to average over the fading



33

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003 2753

where

(6)

and where we have set

(7)

Let be the number of distinct nonzero eigenvalues and assume a re-labeling of the indexes under which , are nonzeroand distinct. Letting denote the multiplicity of as a factorof we can rewrite

(8)

A. The Unknown Channel CaseIn [4], [5], Hochwald et al. show that the PEP under the assumption

of unitary modulation, i.e., and an unknown channelis given by

(9)

where , and where denotes this time, the set ofeigenvalues of . The similarity of the two expres-sions (5) and (9) allows the known and unknown channel cases to betreated simultaneously.

B. The Closed-Form ExpressionFor the special case when all the eigenvalues of are equal,

i.e., , a closed-form expression for theintegral in (5) can be obtained from [6]

(10)

(11)

where

The equal-eigenvalue case applies, for instance, in the situation whenthe collection of signal matrices is drawn from an orthogonal design,see [7].To obtain a closed-form expression for the general, unequal eigen-

value case, we begin with the partial-fraction expansion as in [8]

(12)

where

(13)

(14)

Setting in this expansion, we find that

(15)

Combining (5), (11), (12), and (15), and setting

we obtain the following, general, closed-form expression for the PEP

(16)

where

(17)

After the initial writing of this correspondence, we learned that theexact PEP has already been considered by Simon [9]. However, onlythe case when the difference matrix is orthogonal is consideredthere. Other closed-form expressions for the PEP that are based on thetheory of quadratic forms of Gaussian random variables and which leadto contour integral expressions may be found in [10]–[12].

C. Optimality of Equal EigenvaluesLet denote the set of nonzero eigenvalues and set

. The integer , defined as the number of nonzero eigenvalues, is also known as the transmit diversity. By making use of the arith-

metic-mean, geometric-mean inequality, we have that

(18)

Since the arithmetic and geometric means coincide only when all theterms are equal, it follows from (18) that for a given eigenvalue sum

, the PEP is minimized when all the nonzero eigenvaluesare equal.1

The eigenvalue sum is given by the Euclidean-distance separation be-tween the signal pair

For the particular case of binary phase-shift keying (BPSK) modula-tion, where the components of the signal matrices are drawn from

1While this is a commonly accepted design rule and evidence for this is pro-vided for instance in [13], [1], we have not previously come across a formalproof of this statement.

2752 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

Remarks on Space-Time Codes Including a New LowerBound and an Improved Code

Hsiao-feng Lu, Student Member, IEEE,Yuankai Wang, Student Member, IEEE, P. Vijay Kumar, Fellow, IEEE,

and Keith M. Chugg, Member, IEEE

Abstract—This correspondence presents a new asymptotically exactlower bound on pairwise error probability of a space–time code as wellas an example code that outperforms the comparable orthogonal-de-sign-based space–time (ODST) code. Also contained in the correspondenceare an exact expression for pairwise error probability (PEP), signal designguidelines, and some observations relating to the reception of ODST codes.Index Terms—Orthogonal design, pairwise error probability (PEP),

space-time codes.

I. INTRODUCTION

Consider a space–time coded system with transmit antennas andreceive antennas. We assume under the quasi-static Rayleigh fading

assumption that the channel is fixed for a duration of symbol trans-missions. Let denote the signal alphabet (constellation) and

be a space–time code. Each element of the space–time code isthus a matrix. Given that is the transmitted codeword(code matrix), the received signal is given by

(1)

where , with being the signal-to-noise ratio (SNR). Thecomponents of the noise matrix and thechannel fading-coefficient matrix , respectively, are independentand identically distributed, zero-mean, complex Gaussian randomvariables having common density function

(2)

In order to enable to be considered as the SNR, we constrain thecomponents of the signal matrix to satisfy

for

The known channel case corresponds to the case when the receiverhas perfect channel state information (CSI), i.e., the case when the en-tries of are known to the receiver. As shown in [1], [2], for any pair ofcodewords , the squared Euclidean distance between thesignal components of the corresponding received matrices is givenby

where are the eigenvalues of , whereis the difference-signal matrix, is the corresponding eigenvector ma-trix, and . The authors of [2] also make use of the Cher-

Manuscript received October 30, 2002; revised May 2, 2003. This work wassupported by the National Science Foundation under Grant CCR-0082987. Thematerial in this correspondence was presented in part at the IEEE InternationalSymposium on Information Theory, Lausanne, Switzerland, June/July 2002.The authors are with the Department of Electrical Engineering–Systems, Uni-

versity of Southern California, Los Angeles, CA 90089-2565 USA (e-mail:[email protected]; [email protected]; [email protected]; [email protected] ).Communicated by T. L. Marzetta, Guest Editor.Digital Object Identifier 10.1109/TIT.2003.817475

noff bound to derive the upper bound on the pairwise error probability(PEP) given in (3) for the known channel case

(3)

Throughout the correspondence, wewill use PEP to denote the pairwiseerror probability of codewords.In the first part of the present correspondence, the PEP of a

space–time code over a quasi-static channel is studied, using anapproach that allows both known and unknown channel cases to beconsidered simultaneously. The approach uses Craig’s formula forthe Gaussian probability integral to derive a general, closed-formexpression for the PEP. It is next proven that given a constraint oneigenvalue sum, the optimal signal design is one in whichhas equal eigenvalues. This is followed by an asymptotically exactlower bound on PEP.Some code design principles are stated and used to prove that a

computer-generated signal design introduced here for the case of fourtransmit and four receive antennas significantly outperforms, the cor-responding orthogonal-design-based space–time (ODST) code at allvalues of SNR. Finally, it is shown that ODST codes represent an in-stance of orthogonal signaling, thus providing a clearer explanation forthe simplicity of the ODST code receiver.The exact expression for PEP of codewords appears in Section II

as well as optimality of the equal eigenvalue case. The lower boundon PEP is presented in Section III. Section IV presents signal designguidelines and then goes on to introduce and compare a computer-gen-erated signal design against the corresponding ODST code. The finalsection, Section V, provides a new explanation for the simplicity of theODST code receiver.

II. AN EXACT EXPRESSION FOR THE PEP

We begin with an exact expression for the PEP. By making use ofCraig’s formula [3] for the Gaussian probability integral

(4)

the PEP

provided in [1], [2] can be rewritten in the form

Averaging over all Rayleigh channel realizations, leads to

(5)

0018-9448/03$17.00 © 2003 IEEE

Craig form of Q-function






space-time codes.

I. INTRODUCTION




(1)


(2)


for






(3)







(4)

the PEP



(5)

0018-9448/03$17.00 © 2003 IEEE






space-time codes.

I. INTRODUCTION




(1)


(2)


for






(3)







(4)

the PEP



(5)

0018-9448/03$17.00 © 2003 IEEE


the constellation we have that

(19)

where is the Hamming-distance metric. In the case of quaternaryphase-shift keying (QPSK) modulation where ,

, we have that , where is the Lee-distance metric [14] given by

III. A LOWER BOUND ON PEP WITH THE CORRECTHIGH-SNR ASYMPTOTE

From the alternative expression for PEP given in (5), we obtain asimple lower bound by replacing in the denominator by toobtain

(20)

where

is a constant.Replacing by in (5), one obtains the previously known upper

bound appearing in (3)

(21)

Equality in (21) holds if and only if .Thus, as a summary we have

(22)

The upper and lower bounds in (22) are seen to differ in ratio bythe constant . Values of are tabulated as shown in Table I for somevalues of the product .Fig. 1 compares exact (11), upper (21), and lower (20) bounds on the

PEP versus normalized SNR for the case of equal eigen-values (i.e., for the case , all ) and and showsthe tightness of the lower bound. Note that in the high-SNR region, thelower bound is a better approximation. This is explained later.A power-series expansion for the PEP that is valid for sufficiently

large SNR can be derived as follows:

TABLE IVALUES OF FOR VARIOUS

Fig. 1. Comparison between exact, upper, and lower bounds on the PEP when.

(23)

where is the geometric mean

Comparing (23) with our earlier lower bound on PEP (20), i.e.,

same diversity/fading equation as before, but now D and the branch

powers depend on signal differences



34

These are similar to fading channel code design metrics for single-input/single-output channels

TAROKH et al.: SPACE–TIME CODES FOR HIGH DATA RATE WIRELESS COMMUNICATION 749

Since is unitary, is an orthonormalbasis of C and are independent complex Gaussianrandom variables with variance per dimension and mean

. Let . Thus areindependent Rician distributions with pdf

for , where is the zero-order modified Besselfunction of the first kind.Thus to compute an upper bound on the average probability

of error, we simply average

with respect to independent Rician distributions of toarrive at

(8)

We next examine some special cases.The Case of Rayleigh Fading: In this case, and

as a fortiori for all and . Then the inequality (8)can be written as

(9)

Let denote the rank of matrix , then the kernel of hasdimension and exactly eigenvalues of are zero.Say the nonzero eigenvalues of are , then itfollows from inequality (9) that

(10)

Thus a diversity advantage of and a coding advantageof is achieved. Recall that is theabsolute value of the sum of determinants of all the principal

cofactors of . Moreover, it is easy to see that the ranksof , and are equal.Remark: We note that the diversity advantage is the power

of SNR in the denominator of the expression for the pairwiseerror probability derived above. The coding advantage is anapproximate measure of the gain over an uncoded systemoperating with the same diversity advantage.Thus from the above analysis, we arrive at the following

design criterion.

Design Criteria for Rayleigh Space–Time Codes:• The Rank Criterion: In order to achieve the maximumdiversity , the matrix has to be full rank forany codewords and . If has minimum rankover the set of two tuples of distinct codewords, then

a diversity of is achieved. This criterion was alsoderived in [15].

• The Determinant Criterion: Suppose that a diversity ben-efit of is our target. The minimum of th roots of thesum of determinants of all principal cofactors of

taken over all pairs of distinctcodewords and corresponds to the coding advantage,where is the rank of . Special attention in thedesign must be paid to this quantity for any codewordsand . The design target is making this sum as large aspossible. If a diversity of is the design target, thenthe minimum of the determinant of taken over allpairs of distinct codewords and must be maximized.

We next study the behavior of the right-hand side of inequality(8) for large signal-to-noise ratios. At sufficiently high signal-to-noise ratios, one can approximate the right-hand side ofinequality (8) by

(11)

Thus a diversity of and a coding advantage of

is achieved. Thus the following design criteria is valid for theRician space–time codes for large signal-to-noise ratios.Design Criteria for The Rician Space–Time Codes:• The Rank Criterion: This criterion is the same as thatgiven for the Rayleigh channel.

• The Coding Advantage Criterion: Let denote thesum of all the determinants of principal cofactorsof , where is the rank of . The minimumof the products

taken over distinct codewords and has to be maxi-mized.Note that one could still use the coding advantage

criterion, since the performance will be at least as goodas the right-hand side of inequality (9).

C. The Case of Dependent Fade CoefficientsIn this subsection, we assume that the coefficients are

samples of possibly dependent zero-mean complex Gaussianrandom variables having variance per dimension. This isthe Rayleigh fading, but the extension to the Rician case isstraightforward.

TAROKH et al.: SPACE–TIME CODES FOR HIGH DATA RATE WIRELESS COMMUNICATION 749

Since is unitary, is an orthonormalbasis of C and are independent complex Gaussianrandom variables with variance per dimension and mean

. Let . Thus areindependent Rician distributions with pdf

for , where is the zero-order modified Besselfunction of the first kind.Thus to compute an upper bound on the average probability

of error, we simply average

with respect to independent Rician distributions of toarrive at

(8)

We next examine some special cases.The Case of Rayleigh Fading: In this case, and

as a fortiori for all and . Then the inequality (8)can be written as

(9)

Let denote the rank of matrix , then the kernel of hasdimension and exactly eigenvalues of are zero.Say the nonzero eigenvalues of are , then itfollows from inequality (9) that

(10)

Thus a diversity advantage of and a coding advantageof is achieved. Recall that is theabsolute value of the sum of determinants of all the principal

cofactors of . Moreover, it is easy to see that the ranksof , and are equal.Remark: We note that the diversity advantage is the power

of SNR in the denominator of the expression for the pairwiseerror probability derived above. The coding advantage is anapproximate measure of the gain over an uncoded systemoperating with the same diversity advantage.Thus from the above analysis, we arrive at the following

design criterion.

Design Criteria for Rayleigh Space–Time Codes:• The Rank Criterion: In order to achieve the maximumdiversity , the matrix has to be full rank forany codewords and . If has minimum rankover the set of two tuples of distinct codewords, then

a diversity of is achieved. This criterion was alsoderived in [15].

• The Determinant Criterion: Suppose that a diversity ben-efit of is our target. The minimum of th roots of thesum of determinants of all principal cofactors of

taken over all pairs of distinctcodewords and corresponds to the coding advantage,where is the rank of . Special attention in thedesign must be paid to this quantity for any codewordsand . The design target is making this sum as large aspossible. If a diversity of is the design target, thenthe minimum of the determinant of taken over allpairs of distinct codewords and must be maximized.

We next study the behavior of the right-hand side of inequality(8) for large signal-to-noise ratios. At sufficiently high signal-to-noise ratios, one can approximate the right-hand side ofinequality (8) by

(11)

Thus a diversity of and a coding advantage of

is achieved. Thus the following design criteria is valid for theRician space–time codes for large signal-to-noise ratios.Design Criteria for The Rician Space–Time Codes:• The Rank Criterion: This criterion is the same as thatgiven for the Rayleigh channel.

• The Coding Advantage Criterion: Let denote thesum of all the determinants of principal cofactorsof , where is the rank of . The minimumof the products

taken over distinct codewords and has to be maxi-mized.Note that one could still use the coding advantage

criterion, since the performance will be at least as goodas the right-hand side of inequality (9).

C. The Case of Dependent Fade CoefficientsIn this subsection, we assume that the coefficients are

samples of possibly dependent zero-mean complex Gaussianrandom variables having variance per dimension. This isthe Rayleigh fading, but the extension to the Rician case isstraightforward.


Ortogonal STCs

35

Full diversity, full rate STC with very simple decoding

X =

2

4 s1 s2

�s⇤2 s⇤1

3

5

zk = Hxk +wk k = 0, 1, 2, . . . L� 1

Z = HX+W

minx2C

kZ�HXk2

PPW (Xi,Xj |H) = Q

0

@s

d2(i, j)

2N0

1

A


kH(Xi �Xj)k2

ALAMOUTI: SIMPLE TRANSMIT DIVERSITY TECHNIQUE FOR WIRELESS COMMUNICATIONS 1455

Fig. 4. The BER performance comparison of coherent BPSK with MRRC and two-branch transmit diversity in Rayleigh fading.

likelihood detector:

(15)

Substituting the appropriate equations we have

(16)

These combined signals are then sent to the maximum like-lihood decoder which for signal uses the decision criteriaexpressed in (17) or (18) for PSK signals.

Choose iff

(17)

Choose iff

(18)

Similarly, for using the decision rule is to choose signaliff

(19)

or, for PSK signals,

choose iff(20)

The combined signals in (16) are equivalent to that of four-branch MRRC, not shown in the paper. Therefore, the resultingdiversity order from the new two-branch transmit diversity

scheme with two receivers is equal to that of the four-branchMRRC scheme.It is interesting to note that the combined signals from the

two receive antennas are the simple addition of the combinedsignals from each receive antenna, i.e., the combining schemeis identical to the case with a single receive antenna. Wemay hence conclude that, using two transmit and receiveantennas, we can use the combiner for each receive antennaand then simply add the combined signals from all the receiveantennas to obtain the same diversity order as -branchMRRC. In other words, using two antennas at the transmitter,the scheme doubles the diversity order of systems with onetransmit and multiple receive antennas.An interesting configuration may be to employ two antennas

at each side of the link, with a transmitter and receiver chainconnected to each antenna to obtain a diversity order of fourat both sides of the link.

IV. ERROR PERFORMANCE SIMULATIONS

The diversity gain is a function of many parameters, includ-ing the modulation scheme and FEC coding. Fig. 4 shows theBER performance of uncoded coherent BPSK for MRRC andthe new transmit diversity scheme in Rayleigh fading.It is assumed that the total transmit power from the two

antennas for the new scheme is the same as the transmit powerfrom the single transmit antenna for MRRC. It is also assumedthat the amplitudes of fading from each transmit antennato each receive antenna are mutually uncorrelated Rayleighdistributed and that the average signal powers at each receiveantenna from each transmit antenna are the same. Further, weassume that the receiver has perfect knowledge of the channel.Although the assumptions in the simulations may seem

highly unrealistic, they provide reference performance curvesfor comparison with known techniques. An important issue is


Space Time Multiplexing

36

In STC development, the best one targets is “full rate (rate 1)” — i.e., if the channel is used L times with M-ary constellation, then there should be M*L STC code matrices

In ST-MUX, we send an M-ary signal point out of each TX antenna at each time — these are “rate Nt” under the STC rate definition

3306 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007

Capacity for Suboptimal Receivers forCoded Multiple-Input Multiple-Output Systems

Pansop Kim and Keith M. Chugg, Member, IEEE

Abstract— The optimal detector for coded multiple antennasystems is too complex to be implemented. Even approximationsto the optimal decoder, such as iterating between detection anddecoding, are too complex for many practical systems. Thus, weconsider a number of spatial stream decouplers with the assump-tion of separate decoding and decoupling. The linear zero-forcingdecoupler, the linear minimum mean-squared error (LMMSE)decoupler, the successive soft/hard interference canceller, and theparallel soft/hard interference canceller are investigated. Thecapacity of the channel including these constrained receiverstructures is analyzed. It is shown that no other decouplercan achieve larger capacity than the LMMSE decoupler in asense that it achieves the maximum sub-channel capacities. Thiscontrasts the general belief that interference cancellation schemesare better than linear filters. This is because this general beliefis based on the assumption of perfect interference cancellation,which, in practice, requires joint decoding/decoupling. We discussthe likelihood value derivation from the decouplers, which canbe used for decoding error correction codes. The performance ofparallel transmission of turbo-coded symbols is given to supportthe constrained capacity analysis.

Index Terms— MIMO, equalization, spatial multiplexing, ca-pacity, LMMSE, BLAST, wireless communication.

I. INTRODUCTION

THE capacity of multiple-input multiple-output (MIMO)systems is known to linearly increase with the minimum

number of transmit and receive antennas when the chan-nel coefficients are mutually independent complex circularGaussian random variables [1], [2]. High transmission ratescan be achieved by transmitting parallel streams of data,but the optimum decoding is prohibitively complex since thestreams are coupled through the channel. This has motivatedthe research on suboptimal detectors for MIMO systems.Proposed solutions are spatial linear filters, such as the linearzero-forcing equalizer (LZFE) and the linear minimum mean-squared error (LMMSE) equalizer, and non-linear filters suchas the decision-feedback equalizer (DFE). A Bell-labs LayeredSpace-Time (BLAST) receiver was suggested in [3], [4].Since this is a type of DFE, error propagation can degradethe performance. To minimize this error propagation effect,several ordering methods have been suggested [5], [6], [7].

Manuscript received January 20, 2006; revised August 16, 2006; acceptedAugust 16, 2006. The associate editor coordinating the review of this paperand approving it for publication was D. Huang. The material in this paperwas presented in part at the 2nd International Symopsium on WirelessCommunication Systems, September 2005.

P. Kim is with the System Design Center, Alinks Communiations Incorpo-rated, Torrance, CA 90502 USA (e-mail: [email protected]).

K. M. Chugg is with the Communication Sciences Institute, Univer-sity of Southern California, Los Angeles, CA 90089-2565 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TWC.2007.06051.

It is well known that DFEs, such as the BLAST receiver,perform better than the LMMSE receiver for uncoded MIMOsystems; however, for coded MIMO systems, there exist fewcomparisons of DFE and LMMSE receiver performance inthe literature. In one such comparison, Sweatman et al. [8]examined the LMMSE and BLAST receivers for repetition-coded MIMO systems with delay diversity [9]. Sweatman etal. showed that although the BLAST receiver outperforms theLMMSE receiver in uncoded MIMO systems, the LMMSEreceiver is not always worse than the BLAST receiver forrepetition-coded MIMO systems.

The capacity of suboptimal detectors was analyzed in[10]. Specifically, the capacity was computed for a receiver-constraint and Gaussian modulation. This capacity is the sumof the individual stream capacities after a suboptimal detectorwhere the interference is considered as additional additivewhite Gaussian noise. The DFE is shown to achieve thecapacity of the channel without a receiver-constraint assumingthat there is no error propagation, i.e., perfect interferencecancellation. This result can be anticipated since it was shownthat the DFE for single-input single-output frequency-selectivechannels achieves the receiver-unconstrained capacity [11].However, these results are based on perfect interference can-cellation, which, in practice, requires joint decoding/detection.The conclusions may be different if error propagation isconsidered without joint decoding/detection. This is especiallytrue for the case of high-order modulations, where hard-decisions used to cancel interference can be unreliable.

The capacity may be approached by iterating betweendecoding and detection. As shown in [12], the performance ofthe LMMSE receiver can be improved by employing iterativedetection (i.e., iterating soft-decisions with a soft-in, soft-outdecoder for the channel code). However, iterative LMMSE isnot considered in this paper due to the associated increase incomplexity. Specifically, for a slow fading channel, the (non-iterative) LMMSE receiver may require only one inversion ofan NT ×NT matrix per frame; whereas, the iterative LMMSErequires NT matrix inversions per iteration, per symbol. Forexample, this increase in complexity has led to non-iterativeLMMSE detection being primarily considered in definingIEEE 802.11n wireless local area network standard [13] whichuses a MIMO orthogonal frequency division multiplexing(OFDM) system.

The parallel transmission of coded symbols is also investi-gated in the literature [14], [12], but most consider only BPSKor QPSK, which is also the common assumption in the relatedfield of coded modulation. Since the main interest in space-time multiplexing is an increase in spectral efficiency, it is

1536-1276/07$25.00 c⃝ 2007 IEEE

KIM and CHUGG: CAPACITY FOR SUBOPTIMAL RECEIVERS FOR CODED MULTIPLE-INPUT MULTIPLE-OUTPUT SYSTEMS 3307

{ } 1

0( )

L

lx l

−

=Channel Code S/P

Modulator

Modulator

c d

TNk source

bitsk/r coded

bits

LT

kN

r m N=

modulated symbolsT LN N×

Fig. 1. Coded multiple-input multiple-output transmitter and suboptimal receiver.

Bit likelihood ( )y lc

ChannelDecoder

( )iL dSymbol

likelihood

( ( ))iL x l

RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods

NT symbollikelihoodseach time

(a)

Decoupler ( )y lSymbol

likelihood

1( ( ))L x l1( )x l!

Bit likelihoodcChannelDecoder

( )iL d

( )TNx l!

Symbollikelihood

Bit likelihood

( ( ))TNL x l RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods

(b)

Fig. 2. Suboptimal decoders.

reasonable to consider high order modulation schemes. Here,we analyze the receiver-constrained capacity of suboptimaldetectors and show that the capacity of the LMMSE receiveris larger than those for other equalizers. We also show howto approach this capacity by simulation. We demonstrate theimportance of appropriate likelihood computation to decodechannel codes for high order modulation.

The system model and channel models are given in SectionII. Next, the structures of the existing decouplers are explainedin Section III. Section IV is devoted to the decoupler capacityanalysis. We derive the receiver-constrained capacities for thedecouplers and show that no other decoupler can achievelarger capacity than the LMMSE receiver. The likelihood valuederivation from decoupled signals is discussed in Section V.To verify this capacity analysis, the simulation results forthe parallel transmission of a turbo code are provided inSection VI. Finally, conclusions are given in Section VII. Thenotations, a, a and A, denote a scalar value, a column vectorand a matrix, respectively throughout this paper.

II. SYSTEM MODEL

Wireless systems with NT transmit antennas and NR re-ceive antennas are considered as shown in Figure 1. In thetransmitter, k source bits, c, are encoded by a channel codewith rate r and the coded bits are serial-to-parallel converted.Each stream of coded bits is modulated and transmitted bymultiple transmit antennas. One modulated symbol corre-sponds to m coded bits, i.e., 2m-ary modulation.

The received signal vector at time l can be expressed as

y(l) = H(l)x(l) +

!NT

ρn(l) (1)

where H(l) is a NR × NT channel gain matrix and n(l) isa NR × 1 noise vector. The vectors, x(l) and y(l) are theNT × 1 transmitted signal vector and the NR × 1 receivedsignal vector, respectively. The elements of H(l) and n(l) areindependent identically distributed (i.i.d.) circularly symmetriccomplex Gaussian (CSCG) random variables with zero meanand unit variance. The average energy of each element in x(l)is assumed to be one for simplicity, and ρ is defined as theaverage signal-to-noise ratio per receive antenna.

We consider two channel models which are a quasi-staticRayleigh fading channel model and a fast Rayleigh fadingchannel model [15]. In the quasi-static Rayleigh fading chan-nel model, the channel gains remain constant during one frame(NL vector symbols), but vary independently among differentframes. Whereas, in fast Rayleigh fading channel model, thechannel gains vary independently symbol by symbol. It isassumed that the channel gains are statistically independentand known only to the receiver for both models.

The optimal decoder for i.i.d. equiprobable source bitsselects the most probable source bits, i.e.,

c = argmaxc

f"{y(l)}NL−1

l=0 | c#

. (2)

However, the complexity of the optimal decoder is prohibitivein general. Figure 2(a) illustrates a suboptimal decoder whichperforms symbol/bit likelihood calculation and channel de-coding separately. Let the vectors, y(l), x(l) and n(l) denotethe l-th column vectors of Y, X and N, respectively. Thecomplexity of the suboptimal decoder of Figure 2(a) is alsoprohibitively complex for high order modulation and large NT

because the decoding complexity exponentially increases withm and NT . We thus consider the decoder illustrated in Figure2(b) which employs a decoupler.



37

In STC development, the best one targets is “full rate (rate 1)” — i.e., if the channel is used L times with M-ary constellation, then there should be M*L STC code matrices

In ST-MUX, we send an M-ary signal point out of each TX antenna at each time — these are “rate Nt” under the STC rate definition


Capacity for Suboptimal Receivers forCoded Multiple-Input Multiple-Output Systems

Pansop Kim and Keith M. Chugg, Member, IEEE

Abstract— The optimal detector for coded multiple antennasystems is too complex to be implemented. Even approximationsto the optimal decoder, such as iterating between detection anddecoding, are too complex for many practical systems. Thus, weconsider a number of spatial stream decouplers with the assump-tion of separate decoding and decoupling. The linear zero-forcingdecoupler, the linear minimum mean-squared error (LMMSE)decoupler, the successive soft/hard interference canceller, and theparallel soft/hard interference canceller are investigated. Thecapacity of the channel including these constrained receiverstructures is analyzed. It is shown that no other decouplercan achieve larger capacity than the LMMSE decoupler in asense that it achieves the maximum sub-channel capacities. Thiscontrasts the general belief that interference cancellation schemesare better than linear filters. This is because this general beliefis based on the assumption of perfect interference cancellation,which, in practice, requires joint decoding/decoupling. We discussthe likelihood value derivation from the decouplers, which canbe used for decoding error correction codes. The performance ofparallel transmission of turbo-coded symbols is given to supportthe constrained capacity analysis.

Index Terms— MIMO, equalization, spatial multiplexing, ca-pacity, LMMSE, BLAST, wireless communication.

I. INTRODUCTION

THE capacity of multiple-input multiple-output (MIMO)systems is known to linearly increase with the minimum

number of transmit and receive antennas when the chan-nel coefficients are mutually independent complex circularGaussian random variables [1], [2]. High transmission ratescan be achieved by transmitting parallel streams of data,but the optimum decoding is prohibitively complex since thestreams are coupled through the channel. This has motivatedthe research on suboptimal detectors for MIMO systems.Proposed solutions are spatial linear filters, such as the linearzero-forcing equalizer (LZFE) and the linear minimum mean-squared error (LMMSE) equalizer, and non-linear filters suchas the decision-feedback equalizer (DFE). A Bell-labs LayeredSpace-Time (BLAST) receiver was suggested in [3], [4].Since this is a type of DFE, error propagation can degradethe performance. To minimize this error propagation effect,several ordering methods have been suggested [5], [6], [7].

Manuscript received January 20, 2006; revised August 16, 2006; acceptedAugust 16, 2006. The associate editor coordinating the review of this paperand approving it for publication was D. Huang. The material in this paperwas presented in part at the 2nd International Symopsium on WirelessCommunication Systems, September 2005.

P. Kim is with the System Design Center, Alinks Communiations Incorpo-rated, Torrance, CA 90502 USA (e-mail: [email protected]).

K. M. Chugg is with the Communication Sciences Institute, Univer-sity of Southern California, Los Angeles, CA 90089-2565 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TWC.2007.06051.

It is well known that DFEs, such as the BLAST receiver,perform better than the LMMSE receiver for uncoded MIMOsystems; however, for coded MIMO systems, there exist fewcomparisons of DFE and LMMSE receiver performance inthe literature. In one such comparison, Sweatman et al. [8]examined the LMMSE and BLAST receivers for repetition-coded MIMO systems with delay diversity [9]. Sweatman etal. showed that although the BLAST receiver outperforms theLMMSE receiver in uncoded MIMO systems, the LMMSEreceiver is not always worse than the BLAST receiver forrepetition-coded MIMO systems.

The capacity of suboptimal detectors was analyzed in[10]. Specifically, the capacity was computed for a receiver-constraint and Gaussian modulation. This capacity is the sumof the individual stream capacities after a suboptimal detectorwhere the interference is considered as additional additivewhite Gaussian noise. The DFE is shown to achieve thecapacity of the channel without a receiver-constraint assumingthat there is no error propagation, i.e., perfect interferencecancellation. This result can be anticipated since it was shownthat the DFE for single-input single-output frequency-selectivechannels achieves the receiver-unconstrained capacity [11].However, these results are based on perfect interference can-cellation, which, in practice, requires joint decoding/detection.The conclusions may be different if error propagation isconsidered without joint decoding/detection. This is especiallytrue for the case of high-order modulations, where hard-decisions used to cancel interference can be unreliable.

The capacity may be approached by iterating betweendecoding and detection. As shown in [12], the performance ofthe LMMSE receiver can be improved by employing iterativedetection (i.e., iterating soft-decisions with a soft-in, soft-outdecoder for the channel code). However, iterative LMMSE isnot considered in this paper due to the associated increase incomplexity. Specifically, for a slow fading channel, the (non-iterative) LMMSE receiver may require only one inversion ofan NT ×NT matrix per frame; whereas, the iterative LMMSErequires NT matrix inversions per iteration, per symbol. Forexample, this increase in complexity has led to non-iterativeLMMSE detection being primarily considered in definingIEEE 802.11n wireless local area network standard [13] whichuses a MIMO orthogonal frequency division multiplexing(OFDM) system.

The parallel transmission of coded symbols is also investi-gated in the literature [14], [12], but most consider only BPSKor QPSK, which is also the common assumption in the relatedfield of coded modulation. Since the main interest in space-time multiplexing is an increase in spectral efficiency, it is

1536-1276/07$25.00 c⃝ 2007 IEEE


{ } 1

0( )

L

lx l

−

=Channel Code S/P

Modulator

Modulator

c d

TNk source

bitsk/r coded

bits

LT

kN

r m N=




ChannelDecoder

( )iL dSymbol

likelihood

( ( ))iL x l

RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods


(a)


likelihood

1( ( ))L x l1( )x l!


( )iL d

( )TNx l!

Symbollikelihood

Bit likelihood

( ( ))TNL x l RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods

(b)




II. SYSTEM MODEL



y(l) = H(l)x(l) +

!NT

ρn(l) (1)




c = argmaxc

f"{y(l)}NL−1

l=0 | c#

. (2)





38

this is SO-demod


{ } 1

0( )

L

lx l

−

=Channel Code S/P

Modulator

Modulator

c d

TNk source

bitsk/r coded

bits

LT

kN

r m N=




ChannelDecoder

( )iL dSymbol

likelihood

( ( ))iL x l

RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods


(a)


likelihood

1( ( ))L x l1( )x l!


( )iL d

( )TNx l!

Symbollikelihood

Bit likelihood

( ( ))TNL x l RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods

(b)




II. SYSTEM MODEL



y(l) = H(l)x(l) +

!NT

ρn(l) (1)




c = argmaxc

f"{y(l)}NL−1

l=0 | c#

. (2)




{ } 1

0( )

L

lx l

−

=Channel Code S/P

Modulator

Modulator

c d

TNk source

bitsk/r coded

bits

LT

kN

r m N=




ChannelDecoder

( )iL dSymbol

likelihood

( ( ))iL x l

RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods


(a)


likelihood

1( ( ))L x l1( )x l!


( )iL d

( )TNx l!

Symbollikelihood

Bit likelihood

( ( ))TNL x l RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods

(b)




II. SYSTEM MODEL



y(l) = H(l)x(l) +

!NT

ρn(l) (1)




c = argmaxc

f"{y(l)}NL−1

l=0 | c#

. (2)



sub-optimal, linear (stream) decoupler

Linear Minimum Mean Square Error (LMMSE) decoupler is the most widely used



39


{ } 1

0( )

L

lx l

−

=Channel Code S/P

Modulator

Modulator

c d

TNk source

bitsk/r coded

bits

LT

kN

r m N=




ChannelDecoder

( )iL dSymbol

likelihood

( ( ))iL x l

RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods


(a)


likelihood

1( ( ))L x l1( )x l!


( )iL d

( )TNx l!

Symbollikelihood

Bit likelihood

( ( ))TNL x l RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods

(b)




II. SYSTEM MODEL



y(l) = H(l)x(l) +

!NT

ρn(l) (1)




c = argmaxc

f"{y(l)}NL−1

l=0 | c#

. (2)



Linear Minimum Mean Square Error (LMMSE) decoupler is the most widely used


LMMSE yChoose

minimumMSE

x%Hard

InterferenceCancellation

LMMSEfor remaining

signals

RNTNRN1TN −

)1(~

gx

Fig. 3. BLAST receiver scheme.

Definition 1: A (spatial stream) decoupler is the modulethat estimates xi(l) from the observation y(l) without usingany information about the channel coding, where xi(l) is thei-th element of x(l).

All the processing tasks required for interference cancella-tion receivers (e.g. BLAST) also depend only on y(l). Thus,by this decoupler definition, the ZF receiver, the LMMSEreceiver, the BLAST receiver, the successive soft interferencecanceller (SSIC), the parallel soft/hard interference canceller(PSIC/PHIC) are all decouplers considered in this paper.

The receiver operation of Figure 2(b) is as follows. First, thereceived signal, y(l) is decoupled to produce x(l). From xi(l)which is i-th element of x(l), the symbol likelihood, L(xi(l))is calculated. Soft decisions or the coded bits, {L(d)}, are cal-culated from {L(xi(l))}. Specifically, the soft decisions for them coded bits mapped to xi(l) are computed by marginalizing{L(xi(l))} as described in detail later. The above are examplesof receivers that separate the spatial streams at the receiver byprocessing the received vector for one time instant to estimateeach of the stream symbols. Subsequent processing, exceptfor channel decoding, is done independently on of the streamestimates.

III. DECOUPLER SCHEMES

In this section, we explain the structures of existing decou-plers. Since a decoupler computes {xi(l)}NT

i=1 from only y(l),we drop the dependence on the time index l in the following.

A. LZF, LMMSE decouplers

Let AH be a spatial linear filter, then the decoupled vector,x becomes

x = AHy (3)

where (·)H is the complex conjugate and transpose of a vectoror matrix. For the LZF decoupler, AH is a pseudo-inverse ofH, i.e.,

AH =!HHH

"−1HH , (4)

Note that HHH is invertible with probability one1 if NR is notsmaller than NT . For the LMMSE decoupler, AH minimizesthe mean-squared error, which is E{|x − AHy|2}. Then,

AH =#HHH +

NT

ρI$−1

HH . (5)

1In practice, matrix inversion may be numerically unstable due to finiteprecision effects. The probability that NT independent streams are supportedmay not be one due to this instability.

B. BLAST decoupler

The BLAST decoupler is the ordered successive hard inter-ference cancellation scheme as shown in Figure 3 [4]. First,the received vector, y is filtered by the LMMSE filter in (5) toproduce x. Then, the symbol that contributes the least to themean-squared error is chosen and hard-estimated - i.e., a harddecision is made for this symbol. Next, interference cancel-lation is performed using the hard-estimated symbol and theresulting vector is LMMSE-filtered again with the remainingsymbols. The above operations are repeated until all symbolsare chosen. On each successive LMMSE computation, it isassumed that the previous interference cancellation is perfectand a filter computation of the form in (5) is performed.

The following is the BLAST receiver algorithm:

• g(i) = [1, 2, · · · , NT ], y(1) = y• for i = 1, 2, · · · , NT

– Ht = (hg(i), · · · , hg(NT ))

– P =%HH

t Ht + NTρ I

&−1

– j ⇐ smallest diagonal entry numberof P

– g(i) ⇔ g(j + i) // Swap g(i) with g(j + i)– aH

g(i) ⇐ j-th row of PHHt

– xg(i) = aHg(i)y

(i)

– xg(i) = SLICE(xg(i))– y(i+1) = y(i) − hg(i)xg(i)

where SLICE(xi) is the nearest value from (xi) in thesignal constellation and hi is the i-th column vector ofH.

C. SSIC decoupler

The BLAST decoupler suffers from error propagation dueto the hard interference cancellation. Ordering the detectionis proposed to mitigate this error propagation effect. Anothermethod to reduce this effect is soft interference cancellation[16]. First, the filtered symbol, x0 is derived by LMMSEfiltering, and h0x0 is cancelled from y. Then, the next symbolis estimated by LMMSE-filtering with the remaining symbols.

The following is the detailed SSIC decoupler algorithm.

• y(1) = y• for i = 1, 2, · · · , NT

– Ht = (hi, · · · , hNT)

– aHi ⇐ the first row of (HH

t Ht +NTρ I)−1HH

t

– xi = aHi y(i)

– y(i+1) = y(i) − hixi


{ } 1

0( )

L

lx l

−

=Channel Code S/P

Modulator

Modulator

c d

TNk source

bitsk/r coded

bits

LT

kN

r m N=




ChannelDecoder

( )iL dSymbol

likelihood

( ( ))iL x l

RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods


(a)


likelihood

1( ( ))L x l1( )x l!


( )iL d

( )TNx l!

Symbollikelihood

Bit likelihood

( ( ))TNL x l RN

received symbols

each timeRN

k decodedbits

k/r bitlikelihoods

(b)




II. SYSTEM MODEL



y(l) = H(l)x(l) +

!NT

ρn(l) (1)




c = argmaxc

f"{y(l)}NL−1

l=0 | c#

. (2)




MIMO Capacity Measures

40

Outage Capacity: code over only one fading channel realization

Ergodic Capacity: code over only many fading channel realizations

C(H) = log

2

⇣det

hI+ (⇢/Nt)HH†

i⌘

Cout

(p) : Pr {C(H) > Cout

(p)} > p

Cergodic

= E {C(H)}

X =

2

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s2

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s⇤1

3

5

zk = Hxk +wk k = 0, 1, 2, . . . L� 1

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PPW (Xi,Xj |H) = Q

0

@s

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2N0

1

A


kH(Xi �Xj)k2


MIMO Capacity Measures

41

Outage Capacity: code over only one fading channel realization


0

5

10

15

20

25

30

10 20 30 40 50

UnconstrainedLMMSESSICPSICZFE

Out

age

capa

city

(bits

/sec

/Hz)

SNR (dB)

Fig. 5. Outage capacity comparison of decouplers (1% outage probability,NT =NR=4).

the SNR is increased, but the differences also converge toabout 16 dB in SNR.

Figure 6 shows the ergodic capacities [15]. As in the caseof outage capacity, the ergodic capacity of the LMMSE andthe PSIC decouplers are largest among the decouplers, but thedifferences of the ergodic capacities are smaller than those ofoutage capacities.

V. LIKELIHOOD CALCULATION

The likelihood of each coded bit is needed to decode thechannel code. Biglieri et al. [14] used the squared Euclidiandistance of each symbol, |xi − xi|2, to decode the convolu-tional code. In this section, we derive the exact likelihood ofeach coded symbol for linear decouplers assuming that theinterference is Gaussian-distributed [17].3 An approximatedlikelihood for non-linear decouplers is also derived. Next, weshow how to calculate bit likelihood from symbol likelihood.

A. Symbol likelihood calculation

For the LMMSE, the SSIC and the BLAST decouplers, theestimated symbol can be expressed as

xi = αixi + ni, (43)

and the likelihood is

f(xi|xi) =1

πRni

exp!− |xi − αixi|2

Rni

". (44)

The negative log-likelihood is

L(xi) = − log f(xi|xi) (45)

=|xi − αixi|2

Rni

+ log (πRni) . (46)

When the LMMSE decoupler is used,

αi = aHi hi (47)

3Gaussian approximation is common due to the asymptotic normality ofthe interference in linear receivers (see [18] and references therein).

0

5

10

15

20

25

30

0 5 10 15 20 25 30

UnconstrainedLMMSESSICPSICZFE

Erg

odic

cap

acity

(bits

/sec

/Hz)

SNR (dB)

Fig. 6. Ergodic capacity comparison of decouplers (NT =NR=4).

andRni =

#

j =i

$$aHi hj

$$2 +NT

ρ|ai|

2 , (48)

where ai is the i-th column vector of A in (5). When the SSICdecoupler is used, αi and Rni are given in (11) and (12).

For BLAST decoupler, the statistics of the hard-estimatedsymbol, x, are difficult to calculate. Therefore, we may derivethe likelihood with two assumptions. One is that there is noerror propagation, i.e., perfect interference cancellation, andthe other is that the xi is equal to xi as in the case of SSICdecoupler. When high order modulation is adopted, the secondassumption is more reasonable since |xi − xi| becomes small.We use the second assumption for simulation because errorrate with the second assumption is smaller than with the firstassumption in the simulation results (not shown in this paper).With the first assumption,

αg(i) = aHg(i)hg(i) (49)

and

Rni =NT#

j=i+1

$$$aHg(i)hg(j)

$$$2

+NT

ρ

$$$ag(i)

$$$2. (50)

With the second assumption, αi and Rni are the same as theSSIC case except for the ordering effect.

For PSIC, the estimated vector is given in (19), whereαi and Rni

are given in (22) and (27), respectively. Thelikelihood becomes

f(yi|xi) = (51)

1πNR |Rni

| exp!−

%y

i− αixi

&HR−1

ni

%y

i− αixi

&",

and the negative log-likelihood becomes

L(xi) = − log f(yi|xi) (52)

=%y

i− αixi

&HR−1

ni

%y

i− αixi

&

+ log'πNR |Rni

|(. (53)

The same assumptions for PHIC can be considered. Withthe assumption of xi = xi, the likelihood of xi is given in(53) with Rni

given in (27).


ST-MUX over Quasi-Static Fading

42


10-4

10-3

10-2

10-1

100

10 15 20 25 30 35 40

BLASTUnordered BLASTLMMSE

Sym

bol E

rror

Rat

e

SNR (dB)

Fig. 7. Uncoded symbol error rates of LMMSE and BLAST (NT =NR=4,16QAM, quasi-static fading).

B. Bit likelihood calculation from symbol likelihood

One transmitted symbol, xi is a function of several codedbits as shown in Figure 1. Assume that m coded bits, d =(d0, · · · , dm−1)t, constitutes one symbol, xi. The likelihoodof dj is

f(x|dj) =!

d:dj

f(x|d0, · · · , dm−1)"

k =j

Pr(dk) (54)

=!

d:dj

f(x|x(d))"

k =j

Pr(dk), (55)

where the index i is omitted without loss of generality andd : dj denotes all d consistent with dj . For equal a-prioriprobabilities, the negative log-likelihood becomes

L(dj) = mind:dj

∗L (x(d)) (56)

≈ mind:dj

L (x(d)) , (57)

where

min∗(a, b) = − ln(e−a + e−b) (58)

= min(a, b) − ln(1 + e−|a−b|). (59)

VI. PARALLEL TRANSMISSION OF RANDOM-LIKE CODE

In this section, we construct a simulation for the paralleltransmission of a rate 1/2 turbo code to verify the outagecapacity analysis. The Turbo code in this simulation consistsof two parallel concatenated recursive systematic 4-state con-volution codes with feed-forward generator sequence 5 (octal)and feedback generator sequence 7 (octal). Non-systematicbits are punctured for the rate to be 1/2.

The source bits (1022 bits/frame) are encoded by a turbocode and interleaved by a random interleaver. The interleavedbits are modulated with 16QAM (quadrature amplitude mod-ulation) and transmitted through multiple antennas (NT =4, NR = 4). The following simulation results except Figure 11are on quasi-static Rayleigh fading channel (i.e., the channelis fixed for each turbo code block and varies independently

10-2

10-1

100

5 10 15 20 25

Outage capacity (LMMSE)LMMSEBLAST

Fram

e E

rror

Rat

e, O

utag

e Pr

obab

ility

SNR (dB)

Fig. 8. Performance of parallel transmission of turbo coded symbols withLMMSE and BLAST using squared Euclidean distance. (NT =NR=4, 8bits/sec/Hz, 1022 information bits/frame, quasi-static fading).

from one turbo coded block to the next). All simulations arebased on min-sum processing [19].

Before considering the effects of frame error rate (FER),we consider one uncoded example to illustrate how othershave concluded that non-linear filters such as the BLASTreceiver are better than the LMMSE receiver. Figure 7 showsthe performance of the LMMSE and the unordered/orderedBLAST receiver for parallel transmission of uncoded 16QAMsymbols with 4 transmit antennas and 4 receive antennas.It is seen that both the unordered and the ordered BLASTreceiver outperform the LMMSE by about 2 dB and 6 dBat 10−2 symbol error rate (SER), respectively. That the or-dered BLAST receiver significantly outperforms the LMMSEreceiver is consistent with the results of Sweatman et al. [8]

Figure 8 shows the performance of the LMMSE and theBLAST receiver when the squared Euclidean distances areused to calculate the negative log-likelihoods for the turbodecoder as in [14]. As in the previous example, the BLASTreceiver outperforms the LMMSE receiver. At FER of 10−2,the BLAST receiver outperforms the LMMSE receiver byapproximately 4 dB. Note that both receivers operate farfrom the LMMSE outage capacity which is optimal decouplercapacity. Specifically, at an FER of 10−2, the BLAST receiveroperates approximately 10 dB away from the LMMSE outagecapacity.

Figure 9 shows the performance of the decouplers withthe Gaussian-approximated likelihood calculation as statedin section V. At an FER of 10−2, the LMMSE receiverin this example outperforms the LMMSE receiver decodedwith squared Euclidean distance by 7 dB (i.e., compare withFigure 8). With this improvement, the LMMSE receiver inthis example performs best among all decouplers. This resultis consistent with Theorem 1. The PSIC receiver performs aswell as the LMMSE receiver and the SSIC receiver performsslightly worse than the LMMSE receiver.

The hard interference cancellation schemes (i.e., the BLASTand the PHIC receivers) perform worse than the correspondingsoft interference cancellation schemes (the SSIC and the PSIC


10-2

10-1

100

10 15 20 25 30

Outage capacity (unconstranied)Outage capcity (LMMSE)Outage capcity (SSIC)Outage capcity (PSIC)LMMSESSICBLASTPSICPHIC

Fram

e E

rror

Rat

e, O

utag

e Pr

obab

ility

SNR (dB)

Fig. 9. Performance of Turbo code with LMMSE and BLAST receiverusing likelihood. (NT =NR=4, 8 bits/sec/Hz, 1022 information bits/frame,quasi-static fading).

receivers). This performance degradation may be due to theerror propagation caused by the unreliable hard decisions. Theperformance of the LMMSE receivers is 3.5 dB away from theLMMSE outage capacity at 10−1 FER and 6 dB away at 10−2

FER. This performance difference may be partially caused bythe modulation constraint and the finite size of the frame. Notethat there is a large difference between the LMMSE perfor-mance and receiver-unconstrained outage capacity suggestingthat iterative detection-based approaches, such as iterativeLMMSE, may provide a large performance improvement [12].These results suggest that the outage capacity analysis is agood tool for anticipating the relative performances of thesesystems.

As mentioned, the BLAST receiver outperforms theLMMSE receiver when uncoded SER is measured. However,when turbo-coded FER is measured and an appropriate like-lihood calculation method is employed, the LMMSE receiverhas the best performance among the considered decouplers.Figure 10 illustrates squared error, |xi − xi|2, histogramsfor the LMMSE and BLAST receivers at an SNR of 20dB. Figure 10 shows that the BLAST receiver increases notonly the probability of small squared error (< 0.05) butalso of large squared error (> 0.9). We attribute this tocorrect interference cancellation reducing the squared errorbut incorrect interference cancellation increasing the squarederror.

Based on the results illustrated in Figure 10, we conjecturethat the BLAST receiver’s larger probability of small squarederror leads to better uncoded SER performance yet worseturbo-coded FER performance. Specifically, given an errorevent, the magnitude of square error does not affect theuncoded SER and the uncoded SER is thus dominated by theprobability of small squared error. However, the magnitudeof squared error may degrade turbo-coded FER performancesince these metrics are combined in the soft decision decodingprocess.

Figure 11 shows the performance of the decouplers withthe Gaussian-approximated likelihood calculation for the fast

10-4

10-3

10-2

10-1

100

0.05 0.25 0.45 0.65 0.85 1.05

BLASTLMMSE

Prob

abili

ty

Squared Error

LMMSE

Fig. 10. Squared error histogram of LMMSE and BLAST (NT =NR=4,SNR=20dB, 16QAM, 512000 symbols, quasi-static fading).

10-2

10-1

100

11.5 12 12.5 13 13.5 14 14.5 15

LMMSESSICBLASTPSICPHIC

Fram

e E

rror

Rat

e

SNR (dB)

Fig. 11. Performance of Turbo code with LMMSE and BLAST receiverusing likelihood. (NT =NR=4, 8 bits/sec/Hz, 1022 information bits/frame,fast fading).

Rayleigh fading channels. Similarly, the LMMSE and thePSIC receivers perform best among the decouplers. Thesesimulation results are in agreement with the predictions ofthe ergodic capacity results shown in Figure 6.

VII. CONCLUSIONS

Outage capacities for various spatial stream decouplers havebeen derived and analyzed. It was shown that the LMMSEreceiver achieves the maximum sub-channel capacity when theinput is i.i.d. CSCG random variables. This is in contrast to thegeneral belief that non-linear filters such as decision-feedbackequalizer perform better than linear filters, which is based onresults for uncoded systems. Decision-feedback receivers mayperform better than LMMSE receiver if uncoded error ratesare measuered, but the LMMSE receiver is the best amongnon-iterative decouplers with good channel codes and correctliklihood calculations.

This analysis was supported by simulations. Symbol like-lihood calculations on both linear and non-linear decouplers

uncoded turbo code


MIMO-OFDM Systems

43

Modern Code: used over all sub-carriers, over multiple OFDM blocks, over all antennas

Each sub-carrier channel looks like the MIMO channel models that we have considered

Ergodic capacity is a better model when the system gets many orders of diversity — i.e., many coherence BWs, many coherence times, many independent spatial fading

modes

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